Vickrey Auction

The rules, in one paragraph

Every bidder simultaneously submits a sealed bid b_i. The auctioneer opens the envelopes, awards the item to the bidder with the highest bid b_(1), and charges that winner a price equal to the second-highest bid b_(2). Everyone else pays nothing. That is the complete specification. Bids are sealed (no one sees anyone else's number before submitting) and one-shot (no back-and-forth, no reraises). Compared to a first-price sealed-bid auction the only change is the payment rule: there, the winner pays their own bid; here, the winner pays the runner-up's.

That single edit transforms the strategic problem. In a first-price auction every dollar you bid is a dollar you might have to pay, so you shade your bid below your true value to leave yourself surplus, and the optimal shading depends on guessing the other bidders' distributions. In a Vickrey auction the dollar you bid never appears on the invoice — what you pay is set by someone else's number. Bidding becomes a pure declaration of how much the item is worth to you.

Why truthful bidding is dominant — a two-case proof

Let v be your private valuation, b your sealed bid, and p the highest bid submitted by any of the other bidders. You don't know p when you choose b, but the surplus you eventually receive is fully determined by the triple (v, b, p):

surplus(v, b, p) = (v - p)   if b > p   (you win, pay p)
                 = 0          if b < p   (you lose, pay nothing)

We compare bidding the truth b = v against any other report. There are two interesting cases.

Case 1: shading down, b < v. Three sub-regions for p.

• p < b < v. Truth wins; shaded bid also wins. Both pay p. Same surplus.
• b < p < v. Truth wins and earns v − p > 0. Shaded bid loses and earns 0. Truth strictly better.
• v < p. Both lose. Same surplus of 0.

Case 2: bidding up, b > v. Three sub-regions again.

• p < v < b. Both win, both pay p, same surplus.
• v < p < b. Truth loses and earns 0. Overbid wins but pays p > v, earning v − p < 0. Truth strictly better.
• b < p. Both lose. Same surplus.

In every case, bidding b = v is at least as good as any alternative, regardless of what the other bidders do. That is the definition of a weakly dominant strategy. Notice the proof uses no probability, no equilibrium reasoning, no assumption about the other bidders' valuations. It works pointwise on p.

Revenue equivalence

The shocking implication of Vickrey's 1961 paper, polished by Roger Myerson in 1981 and Riley-Samuelson the same year, is that under broad conditions the seller is indifferent between Vickrey and the three other standard formats. Specifically, when valuations are independently drawn from a common distribution F, bidders are risk-neutral, the allocation rule gives the item to the bidder with the highest valuation, and the bidder with the lowest possible valuation expects zero surplus, all four mechanisms yield the same expected revenue for the seller — and the same expected surplus for each bidder.

Auction	Format	Equilibrium bid	Winner pays	Expected revenue (n bidders)
English	Open ascending	Stay in up to v	Just above 2nd value	E[V_(n-1)]
Dutch	Open descending	Strategically equivalent to first-price	Own bid	E[V_(n-1)]
First-price sealed-bid	Sealed	Shade to (n−1)/n · v under uniform F	Own (shaded) bid	E[V_(n-1)]
Vickrey (second-price)	Sealed	Bid v truthfully	2nd-highest bid	E[V_(n-1)]

The common expression E[V_(n-1)] is the expected second-order statistic of the valuation distribution. Vickrey's elegance is not that he found a higher-revenue mechanism — he didn't — but that he found the one in which equilibrium reasoning is trivial. Myerson's broader optimal-auction theorem then characterises when a seller can do better than revenue-equivalent, by introducing reserve prices or asymmetric handling.

A four-bidder worked example

Four bidders draw valuations independently from the uniform distribution on [0, 100]. Suppose the realised values are v_1 = 78, v_2 = 64, v_3 = 51, v_4 = 22. We compare what each auction produces.

Auction	What bidder i submits	Bids	Winner	Price paid
Vickrey	b_i = v_i	78, 64, 51, 22	Bidder 1	64
First-price (Nash, uniform)	b_i = (n−1)/n · v_i = 0.75 v_i	58.5, 48, 38.25, 16.5	Bidder 1	58.5
English (button auction)	Stay in to v_i	drop at 22, 51, 64	Bidder 1	~64 (just above)
Dutch	Accept at strategic price	price falls to 58.5	Bidder 1	58.5

Note that in this one realisation Vickrey and English produce ~64 while first-price and Dutch produce 58.5 — they differ at the realised draw. Averaged over the distribution, however, expected revenue is identical. The intuition is that first-price bidders shade more aggressively as competition grows, exactly enough to match the expected second-highest valuation in Vickrey. The variance of the seller's revenue differs across mechanisms even when the mean does not, which is one reason risk-averse sellers may not be indifferent in practice.

Vickrey-Clarke-Groves: the multi-item generalisation

Vickrey's paper also handled the case where the seller has multiple identical items and bidders want at most one. Clarke (1971) and Groves (1973) extended the mechanism to the much more general problem where bidders have arbitrary valuations over bundles of heterogeneous items. The Vickrey-Clarke-Groves (VCG) mechanism asks each bidder to report their full valuation function, then awards items to maximise reported social welfare and charges each winner their externality — the difference between (a) the total welfare the others would have enjoyed without them and (b) the total welfare the others actually enjoy in the chosen allocation. The mechanism is strategy-proof, efficient, and reduces to second-price in the single-item case.

VCG payment for winner i
  p_i = max_{allocation x} Σ_{j≠i} v_j(x)        // welfare others would have
       − Σ_{j≠i} v_j(x*)                  // welfare others actually get

VCG has serious drawbacks that limit deployment. Revenue can collapse to zero in combinatorial settings with substitutable items. The mechanism is non-monotonic — adding a bidder can lower the seller's revenue. It is vulnerable to false-name bidding when identities are cheap (one bidder can register as several). And computing optimal allocations is NP-hard for general combinatorial auctions, so real VCG-inspired systems use heuristic or anytime approximations.

Where Vickrey actually appears

• eBay proxy bidding. Looks like an English auction but operates as a Vickrey auction. Bidders enter a maximum proxy bid; eBay's robot raises in small increments on their behalf, ending the auction at one increment above the runner-up's max. The dominant strategy is to submit your true reservation value as your proxy and not return until the close.
• Google Ads (GSP, Vickrey-flavoured). Search-ad slots are sold by the generalised second-price auction. For a single slot, GSP and Vickrey coincide. With multiple slots, GSP is intuitive but not strictly strategy-proof; Google chose it over true VCG (Clarke 1971) because advertisers find the "you pay just above the next bid" framing easier to trust.
• FCC spectrum auctions. The US Federal Communications Commission has used variants of VCG and the simultaneous-multiple-round-ascending mechanism since 1994, raising over $230 billion through 2024. The 2017 broadcast incentive auction (clearing TV spectrum for mobile) used a Vickrey-style reverse-then-forward design.
• Kidney exchange. The Roth-Sönmez-Ünver pairwise and chain exchange algorithms use deferred-acceptance variants that inherit the strategy-proofness of Vickrey-style designs, ensuring patients and donors can report their priorities truthfully.
• Treasury bill auctions. The US Treasury switched to a uniform-price (single-price) auction for marketable securities in 1992, which has Vickrey-like incentive properties at the margin. The full multi-unit Vickrey would have been theoretically preferable but operationally complex.
• Adobe and Microsoft real-time bidding. Display-ad exchanges (OpenRTB, Google Ad Exchange) moved to a first-price model in 2019 after a long Vickrey era, citing transparency and arbitrage between exchanges. The transition was a rare reversal of a Vickrey-favoured industry.

Why Vickrey is rare in practice

Despite forty years of theory, Vickrey auctions are conspicuously uncommon outside the proxy-disguised forms. The reasons are practical, not theoretical.

• Seller credibility. Sellers find the rule unintuitive and bidders worry the seller will fabricate a phantom second bid to push the price up. In a public English auction this fraud is harder; in a sealed Vickrey it requires only that the seller add an extra envelope. Trust is fragile.
• Information leakage. The winner learns the runner-up's exact bid. In repeated auctions or in adjacent markets that fact is competitively useful, and reluctance to broadcast one's true valuation can chill participation.
• Collusion vulnerability. A cartel can submit one high bid and many trivial bids; the price ends up being one of theirs. The same trick is harder in an English auction where competing buyers see whether anyone challenges the leader.
• Multi-item revenue. VCG generalisations can extract very low or zero revenue when items are substitutes, and the seller's revenue is not monotone in the number of participants. For revenue-conscious sellers this is intolerable.
• Behavioural deviations. Laboratory experiments routinely find systematic overbidding in second-price auctions: bidders bid above their valuations on average. The dominant strategy is dominant only for fully rational bidders, and overbidding suggests something subtle about regret, fear of losing, or misunderstanding the rule.

Variants and extensions

• Reserve price. Setting a reserve r excludes bids below r; the winner pays max(b_(2), r). This violates the unconditional revenue-equivalence assumption (lowest-valuation bidder no longer has zero expected surplus) but raises expected revenue when v < r occurs with non-trivial probability. Myerson 1981 characterises the optimal reserve.
• Generalised second-price (GSP). Used by Google and Microsoft search ads. Slot k goes to bidder ranked k by bid; bidder k pays the bid of bidder k+1. Strategy-proofness fails with k > 1, but locally truthful equilibria exist and the mechanism is operationally simple.
• k-th price auction. Winner pays the k-th highest bid. k = 1 is first-price; k = 2 is Vickrey; k ≥ 3 are studied theoretically but rare in practice.
• All-pay Vickrey (war of attrition). Everyone pays their bid, winner gets the item. Strategy-proofness fails; equilibrium bidding is below value. Models lobbying, R&D races, and rent-seeking contests.
• Vickrey with allocative externalities. When losing matters (a rival winning hurts you), the simple second-price rule no longer yields a dominant strategy and Myerson-style optimal mechanisms become asymmetric.

Common pitfalls

• Confusing Vickrey with English. They are revenue-equivalent in the standard environment but mechanically different. English is open and dynamic; Vickrey is sealed and simultaneous. Information revelation differs sharply when bidders' valuations have a common component or affiliation.
• Assuming GSP is strategy-proof. Generalised second-price is not VCG and not DSIC for more than one slot. The locally envy-free equilibria of Edelman-Ostrovsky-Schwarz approximate Vickrey behaviour, but ignoring the gap leads to wrong strategy advice for advertisers.
• Ignoring the seller's incentive to cheat. The Vickrey payment rule depends on the seller honestly reporting the second-highest bid. Lab experiments and field deployments have caught sellers padding the second bid. Without transparent escrow or a trusted auctioneer the dominance argument breaks.
• Applying revenue equivalence past its assumptions. The theorem fails for risk-averse bidders, correlated valuations, asymmetric distributions, budget constraints, or risk-loving sellers. Each generalisation has its own optimal mechanism — usually NOT Vickrey.
• Treating VCG revenue as a free lunch. Strategy-proofness in multi-item VCG comes at the cost of revenue. Real auctioneers compromise — Australia's spectrum sales use a "core-selecting" payment rule that gives up some strategy-proofness for better revenue.